Mathematical Methods XIA

Physical Chemistry: An Advanced Treatise: Mathematical Methods, Volume XIA, is devoted to mathematical techniques of interest to chemists. The purpose of this treatise is to present a comprehensive treatment of physical chemistry for advanced students and investigators in a reasonably small number of volumes. An attempt has been made to include all important topics in physical chemistry together with borderline subjects which are of particular interest and importance.
The book begins with discussions of elementary concepts such as linear vector spaces; generalized function theory; complex variable theory; boundary-value problems; approximating functions and their applications in numerical differentiation, integration, and the solution of differential equations; and group theory. These are followed by more advanced and specialized chapters that emphasize chemical applications rather than mathematical rigor.
This book provides the student of physical chemistry with a basic understanding of those additional mathematical techniques which are important in chemistry and should enable him to read the current literature in theoretical chemistry.

Contenu

List of Contributors
Foreword
Preface
Contents of Previous and Future Volumes
Chapter 1 / Linear Vector Spaces
I. Introduction: Vectors in the Physical Sciences
II. Linear Vector Spaces
III. Example: Three-Dimensional Euclidean Vectors-I
IV. Vector Transformations
V. Matrices
VI. Example: Three-Dimensional Euclidean Vectors-II
VII. Vector Spaces of Infinite Dimension
References
Chapter 2 / Generalized Functions
I. Introduction
II. Definitions
III. The Algebra of Generalized Functions
IV. The Calculus of Generalized Functions
V. Some Singular Generalized Functions
VI. Fourier Transforms
VII. Laplace Transforms
VIII. Conclusion
References
Chapter 3/Complex Variable Theory
I. Introduction
II. Complex Numbers
III. Analytic Functions of a Complex Variable
IV. Complex Integration
V. Power Series
VI. Elementary Functions
VII. Evaluation of Real Definite Integrals
VIII. Higher Transcendental Functions
IX. On Fourier Transforms
X. Quantum Chemistry Integrals
XI. A Formula of Lagrange and Nondegenerate Perturbation Theory
References
Chapter 4 / Boundary-Value Problems
I. Introduction
II. Some Typical Boundary-Value Problems
III. The D'Alembert Solution of the Wave Equation
IV. Separation of Variables
V. Eigenvalues, Eigenfunctions, and Expansion Problems
VI. Boundary-Value Problems in Cylindrical Coordinates
VII. Boundary-Value Problems in Spherical Coordinates
VIII. Green's Functions
IX. Laplace Transform Methods
X. Conformal Mapping
References
Chapter 5 / Numerical Analysis
I. Introduction
II. Approximation by Polynomial Interpolation
III. Approximation by Spline Interpolation
IV. Approximation by Least Squares
V. Numerical Differentiation
VI. Approximate Integration or Quadrature
VII. Differential Equations
VIII. Equations in a Single Unknown
IX. Systems of Linear Equations
X. Special Methods for Solving Sparse Sets of Equations
Appendix
References
Chapter 6 / Group Theory
I. Introduction
II. Definitions
III. Symmetry Operators
IV. Group Representation Theory
V. Some Applications in Molecular Quantum Mechanics
VI. The Permutation Group and Spin
VII. Continuous Groups
VIII. Group Theory and the Solid State
References
Chapter 7 / Density Matrices
I. Introduction
II. The Full Density Matrix
III. The Reduced Density Matrix
IV. The N-Representability Problem
V. The Single-Particle Reduced Density Matrix
VI. The Second-Order Reduced Density Matrix
VII. General Geminal Wave Functions
VIII. Condensation Phenomena
References
Chapter 8 / The Green's Function Method
I. Introduction
II. Double-Time Temperature-Dependent Green's Functions
III. Spectral Representations
IV. Properties of the Green's Functions
V. The Reaction of a System to an External Perturbation
VI. Calculation of the Green's Functions
VII. Charge-Transfer Spectra of Molecular Crystals
VIII. Perturbation Theory for the Green's Functions
References
Author Index
Subject Index